It could have helped, although running a "blue light" as in my state, is what volunteer folk use. It is a courtesy light to inform other motorists they need to get someone to assist with an emergency. Having one, and using it is a big responsibility that many people who use them, more often than not.. forget. Using one, by no means gives you a right or "free pass" to speed, run stop sign or red lights. You have to follow the traffic laws like everyone else. You don't even have to pull to the side of the road to let them pass you by.
Now, if you want to run a red light, like the ambulances have. That is a whole different story in which I am not very familiar with. I am sure it is not an easy process and not allowed for just everyone, regardless if you are a DR or not. Surgeons and their team members who are on call at trauma centers and other hospitals, don't use them. They have to travel like everyone else, but have to be within certain distances when they are on-call.
Now, for this particular situation. The cop should have taken the man's situation seriously, and asked for credentials. When they are supplied, the cop should have had a heart and taken him to the hospital himself in an emergent manner, so he could assist this woman in crisis. Then given him a ticket for breaking the law. Sorry, he broke the law.. no get out of jail free cards, even for DR.
If this was such a huge ordeal that he had to be there so fast, I am sure there were others with this woman who could have assisted until he got there. Nurses are highly trained in their field and could obtain orders from the DR over the phone, or another DR in the hospital, one of which who could have and should have come to the aid of the woman. Maybe not an obstetrics DR, but still a DR none the less. Just cause you went into a differently specialty does not mean you didn't learn about the others.